Which predictive algorithm is most appropriate for a proceeding situation?
One approach would be to use a Sequence Processing Neural architecture - one option is a recurrent network. These were specifically designed with your intent in mind.

They can consume sequences of data and learn from these in order to predict subsequent time-steps. Though these will require you to understand best practices for implementation.

If your task isn't too complex you could make use of a forward algorithm or some variant. Many of these models make use of simplifying assumptions - if your use case cannot admit these you'll have to go for something more advanced.

If your prediction task is complex and requires heavy use of long term dependencies you may have to go for a more cutting edge architecture like a transformer - but I heard that these are difficult to implement and most definitely require familiarity with deep learning systems. 

Finally, if you need a system that can understand data and dependencies across very large spans of time then you may have to wait for the science to progress.

I hope this helps.